(Requires Calculus)For the simple linear regression model of Chapter 4,

Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }

, the OLS estimator for the intercept was

\hat { \beta } _ { 0 } = \bar { Y } - \hat { \beta } _ { 1 } \bar { X }

, and

\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } - n \overline { X Y } } { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } - n \bar { X } ^ { 2 } }

Intuitively, the OLS estimators for the regression model

Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 i } + \beta _ { 2 } X _ { 2 i } + u _ { i }

might be

\hat { \beta } _ { 0 } = \bar { Y } - \hat { \beta } _ { 1 } \bar { X } _ { 1 } - \hat { \beta } _ { 2 } \bar { X } _ { 2 } , \hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } \bar { X } _ { 1i } Y _ { i } - n \bar { X } _ { 1 } \bar { Y } } { \sum _ { i = 1 } ^ { n } \bar { X } _ { 1 i } ^ { 2 } - n \bar { X } _ { 1 } ^ { 2 } }

and

\hat { \beta } _ { 2 } = \frac { \sum _ { i = 1 } ^ { n } \bar { X } _ { 2 i } Y _ { i } - n \bar { X } _ { 2 } \bar { Y } } { \sum _ { i = 1 } ^ { n } \bar { X } _ { 2 i } ^ { 2 } - n \bar { X } _ { 2 } ^ { 2 } }

By minimizing the prediction mistakes of the regression model with two explanatory variables, show that this cannot be the case.

The OLS Residuals in the Multiple Regression Model

Multiple Choice